Nonlinear Oscillations of Hamiltonian PDEs [electronic resource] / by Massimiliano Berti.
Contributor(s): SpringerLink (Online service)Material type: TextSeries: Progress in Nonlinear Differential Equations and Their Applications: 74Publisher: Boston, MA : Birkhäuser Boston, 2007Description: XIV, 180 p. 10 illus. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9780817646813Subject(s): Mathematics | Approximation theory | Dynamics | Ergodic theory | Partial differential equations | Applied mathematics | Engineering mathematics | Number theory | Physics | Mathematics | Partial Differential Equations | Dynamical Systems and Ergodic Theory | Approximations and Expansions | Number Theory | Applications of Mathematics | Mathematical Methods in PhysicsAdditional physical formats: Printed edition:: No titleDDC classification: 515.353 LOC classification: QA370-380Online resources: Click here to access online
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Finite Dimension -- Infinite Dimension -- A Tutorial in Nash–Moser Theory -- Application to the Nonlinear Wave Equation -- Forced Vibrations.
Many partial differential equations (PDEs) that arise in physics can be viewed as infinite-dimensional Hamiltonian systems. This monograph presents recent existence results of nonlinear oscillations of Hamiltonian PDEs, particularly of periodic solutions for completely resonant nonlinear wave equations. After introducing the reader to classical finite-dimensional dynamical system theory, including the Weinstein–Moser and Fadell–Rabinowitz resonant center theorems, the author develops the analogous theory for completely resonant nonlinear wave equations. Within this theory, both problems of small divisors and infinite bifurcation phenomena occur, requiring the use of Nash–Moser theory as well as minimax variational methods. These techniques are presented in a self-contained manner together with other basic notions of Hamiltonian PDEs and number theory. This text serves as an introduction to research in this fascinating and rapidly growing field. Graduate students and researchers interested in nonlinear variational techniques as well in small divisors problems applied to Hamiltonian PDEs will find inspiration in the book.